Rigidity of Thin Disk Configurations.

Abstract

The main result of this thesis is a rigidity theorem for configurations of closed disks in the plane. More precisely, fix two collections C and C' of closed disks, sharing a contact graph which (mostly-)triangulates the complex plane, so that for all corresponding pairs of intersecting disks Di, Dj in C and Di', Dj' in C', we have that the intersection angle between the boundary circles of Di and Dj equals the analogous angle for Di' and Dj'. We require the extra condition that the collections C and C' are thin, meaning that no pair of disks of C meet in the interior of a third, and similarly for C'. Then C and C' differ by a Euclidean similarity.

Our proof is elementary, using essentially only plane topology arguments and manipulations by Moebius transformations. In particular, we generalize an argument which was previously used by Schramm to prove the rigidity of configurations of pairwise interiorwise disjoint closed disks having contact graphs triangulating the complex plane. It was previously thought that his proofs depended too crucially on the pairwise interiorwise disjointness of the disks for there to be a hope for generalizing them to the setting of configurations of overlapping disks.

Analogous versions of our rigidity theorem have easier proofs via a discrete version of the Maximum Modulus Principle in the case where C and C' share a contact graph which (mostly-)triangulates the hyperbolic plane, or the Riemann sphere. We describe these proofs as well. These are relatively straightforward generalizations of the corresponding proofs in the case of configurations of pairwise interiorwise disjoint disks. Then by a simple argument via covering space theory and the Uniformization Theorem, we get an analogous rigidity statement for thin disk configurations having contact graphs triangulating an arbitrary Riemann surface.

We include a brief and gentle introduction intended for non-mathematicians. Then we give a survey of the field of circle packing, which is the area that our result fits into. We also state some open problems and conjectures from this area, including conjectured generalizations both of our main result and of our main technical theorem.